Eigenvalues, geometry and instability in conservative models in applied mathematics.

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Most of the equations derived to model physical phenomena support some formof special solutions such as standing or traveling waves, or other coherentstructures. These special solutions often correspond to real, observablephysical phenomena in the systems that these model equations are meant torepresent. One important aspect of this modeling is the stability of thesespecial solutions, as the stability of solutions determines if thesesolutions are likely to be physically observed. This project is aimed atunderstanding the stability and instability these coherent structures.This usually amounts to computing the index of the operator found bylinearizing about one of these coherent structures - that is the number ofeigenvalues of positive real part. This counts the dimension of theunstable manifold to the solution, giving valuable information on thedynamics of nearby solutions. While the emphasis of this project is onconservative systems we consider both conservative and dissipative systems.We use analytical, asymptotic and geometric techniques to identify unstableeigenvalues and to count the number of such eigenvalues.There are many mathematical models that are important for science andengineering that support solutions in the form of traveling waves. Oneexample is an equation known as the Korteweg-DeVries equation, an equationwhich governs the behavior of water in a narrow shallow body such as acanal. The Korteweg-DeVries equation has solutions called solitary waveswhich behave just as experience would suggest - they correspond to aquantity of water which propagates along without changing shape. Similarequations govern the propagation of light waves, propagation of adisturbance through a network such as the power grid, etc. It isimportant to understand the extent to which these solutions accuratelymodel the behavior of the underlying system. This is the question ofstability, which measures how robust these special solutions are. If aspecial solution like a traveling wave is stable it means that nearbysolutions behave in a similar way. This means such solutions are robust,and are likely to be observed: if the conditions are not exactly thosenecessary to produce a traveling wave but are close we expect to see asolution which is close to the special one. An unstable solution, on the other hand, is not robust. In order to observe these unstable solutionone must produce exactly the right conditions, which is very difficultto achieve in practice. This means that such solutions are more of mathematical interest than of physical importance. This project isprimarily concerned with developing mathematical techniques to understand the stability of traveling waves in a number of models including the Kuramoto model (a model for the behavior of power networks) and several nonlinear dispersive equations (which govern light waves in nonlinear media, water waves, waves in plasmas and many other phenomena).